Chapter 20

A diesel engine performs 2200 J of mechanical work and discards 4300 J of heat each cycle. (a) How much heat must be supplied to the engine in each cycle? (b) What is the thermal efficiency of the engine?

An aircraft engine takes in 9000 J of heat and discards 6400 J each cycle. (a) What is the mechanical work output of the engine during one cycle? (b) What is the thermal efficiency of the engine?

A gasoline engine takes in 1.61 \(\times\) 10\(^4\) J of heat and delivers 3700 J of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of 4.60 \(\times\) 10\(^4\) J/g. (a) What is the thermal efficiency? (b) How much heat is discarded in each cycle? (c) What mass of fuel is burned in each cycle? (d) If the engine goes through 60.0 cycles per second, what is its power output in kilowatts? In horsepower?

A gasoline engine has a power output of 180 \(kW\)(about 241 hp). Its thermal efficiency is 28.0%. (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

(a) Calculate the theoretical efficiency for an Otto-cycle engine with \(\gamma\) = 1.40 and \(r\) = 9.50. (b) If this engine takes in 10,000 J of heat from burning its fuel, how much heat does it discard to the outside air?

The Otto-cycle engine in a Mercedes-Benz SLK230 has a compression ratio of 8.8. (a) What is the ideal efficiency of the engine? Use \(\gamma\) = 1.40. (b) The engine in a Dodge Viper GT2 has a slightly higher compression ratio of 9.6. How much increase in the ideal efficiency results from this increase in the compression ratio?

A refrigerator has a coefficient of performance of 2.10. In each cycle it absorbs 3.10 \(\times\) 10\(^4\) J of heat from the cold reservoir. (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat is discarded to the high-temperature reservoir?

A freezer has a coefficient of performance of 2.40. The freezer is to convert 1.80 kg of water at 25.0\(^\circ\)C to 1.80 kg of ice at -5.0\(^\circ\)C in one hour. (a) What amount of heat must be removed from the water at 25.0\(^\circ\)C to convert it to ice at -5.0\(^\circ\)C? (b) How much electrical energy is consumed by the freezer during this hour? (c) How much wasted heat is delivered to the room in which the freezer sits?

A Carnot engine is operated between two heat reservoirs at temperatures of 520 \(K\) and 300 \(K\). (a) If the engine receives 6.45 \(kJ\) of heat energy from the reservoir at 520 \(K\) in each cycle, how many joules per cycle does it discard to the reservoir at 300 \(K\)? (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?

A Carnot engine whose high-temperature reservoir is at 620 K takes in 550 J of heat at this temperature in each cycle and gives up 335 J to the low- temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? What is (b) the temperature of the low-temperature reservoir; (c) the thermal efficiency of the cycle?

A Carnot engine has an efficiency of 66% and performs 2.5 \(\times\) 10\(^4\) J of work in each cycle. (a) How much heat does the engine extract from its heat source in each cycle? (b) Suppose the engine exhausts heat at room temperature (20.0\(^\circ\)C). What is the temperature of its heat source?

A certain brand of freezer is advertised to use 730 kW \(\cdot\) h of energy per year. (a) Assuming the freezer operates for 5 hours each day, how much power does it require while operating? (b) If the freezer keeps its interior at -5.0\(^\circ\)C in a 20.0\(^\circ\)C room, what is its theoretical maximum performance coefficient? (c) What is the theoretical maximum amount of ice this freezer could make in an hour, starting with water at 20.0\(^\circ\)C?

A Carnot refrigerator is operated between two heat reservoirs at temperatures of 320 K and 270 K. (a) If in each cycle the refrigerator receives 415 J of heat energy from the reservoir at 270 K, how many joules of heat energy does it deliver to the reservoir at 320 K? (b) If the refrigerator completes 165 cycles each minute, what power input is required to operate it? (c) What is the coefficient of performance of the refrigerator?

A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation, the heat rejected by the engine melts 0.0400 kg of ice. During this time, how much work \(W\) is performed by the engine?

You design an engine that takes in 1.50 \(\times\) 10\(^4\) J of heat at 650 K in each cycle and rejects heat at a temperature of 290 K. The engine completes 240 cycles in 1 minute. What is the theoretical maximum power output of your engine, in horsepower?

A 4.50-kg block of ice at 0.00\(^\circ\)C falls into the ocean and melts. The average temperature of the ocean is 3.50\(^\circ\)C, including all the deep water. By how much does the change of this ice to water at 3.50\(^\circ\)C alter the entropy of the world? Does the entropy increase or decrease? (\(Hint\): Do you think that the ocean temperature will change appreciably as the ice melts?)

You decide to take a nice hot bath but discover that your thoughtless roommate has used up most of the hot water. You fill the tub with 195 kg of 30.0\(^\circ\)C water and attempt to warm it further by pouring in 5.00 kg of boiling water from the stove. (a) Is this a reversible or an irreversible process? Use physical reasoning to explain. (b) Calculate the final temperature of the bath water. (c) Calculate the net change in entropy of the system (bath water + boiling water), assuming no heat exchange with the air or the tub itself.

A 15.0-kg block of ice at 0.0\(^\circ\)C melts to liquid water at 0.0\(^\circ\)C inside a large room at 20.0\(^\circ\)C. Treat the ice and the room as an isolated system, and assume that the room is large enough for its temperature change to be ignored. (a) Is the melting of the ice reversible or irreversible? Explain, using simple physical reasoning without resorting to any equations. (b) Calculate the net entropy change of the system during this process. Explain whether or not this result is consistent with your answer to part (a).

You make tea with 0.250 kg of 85.0\(^\circ\)C water and let it cool to room temperature (20.0\(^\circ\)C). (a) Calculate the entropy change of the water while it cools. (b) The cooling process is essentially isothermal for the air in your kitchen. Calculate the change in entropy of the air while the tea cools, assuming that all of the heat lost by the water goes into the air. What is the total entropy change of the system tea + air?

Three moles of an ideal gas undergo a reversible isothermal compression at 20.0\(^\circ\)C. During this compression, 1850 J of work is done on the gas. What is the change of entropy of the gas?

A box is separated by a partition into two parts of equal volume. The left side of the box contains 500 molecules of nitrogen gas; the right side contains 100 molecules of oxygen gas. The two gases are at the same temperature. The partition is punctured, and equilibrium is eventually attained. Assume that the volume of the box is large enough for each gas to undergo a free expansion and not change temperature. (a) On average, how many molecules of each type will there be in either half of the box? (b) What is the change in entropy of the system when the partition is punctured? (c) What is the probability that the molecules will be found in the same distribution as they were before the partition was punctured- that is, 500 nitrogen molecules in the left half and 100 oxygen molecules in the right half?

A lonely party balloon with a volume of 2.40 \(L\) and containing 0.100 mol of air is left behind to drift in the temporarily uninhabited and depressurized International Space Station. Sunlight coming through a porthole heats and explodes the balloon, causing the air in it to undergo a free expansion into the empty station, whose total volume is 425 m\(^3\). Calculate the entropy change of the air during the expansion.

An ideal Carnot engine operates between 500\(^\circ\)C and 100\(^\circ\)C with a heat input of 250 J per cycle. (a) How much heat is delivered to the cold reservoir in each cycle? (b) What minimum number of cycles is necessary for the engine to lift a 500-kg rock through a height of 100 m?

An average sleeping person metabolizes at a rate of about 80 \(W\) by digesting food or burning fat. Typically, 20% of this energy goes into bodily functions, such as cell repair, pumping blood, and other uses of mechanical energy, while the rest goes to heat. Most people get rid of all this excess heat by transferring it (by conduction and the flow of blood) to the surface of the body, where it is radiated away. The normal internal temperature of the body (where the metabolism takes place) is 37\(^\circ\)C, and the skin is typically 7 C\(^\circ\) cooler. By how much does the person's entropy change per second due to this heat transfer?

CP A certain heat engine operating on a Carnot cycle absorbs 410 J of heat per cycle at its hot reservoir at 135\(^\circ\)C and has a thermal efficiency of 22.0%. (a) How much work does this engine do per cycle? (b) How much heat does the engine waste each cycle? (c) What is the temperature of the cold reservoir? (d) By how much does the engine change the entropy of the world each cycle? (e) What mass of water could this engine pump per cycle from a well 35.0 m deep?

Digesting fat produces 9.3 food calories per gram of fat, and typically 80% of this energy goes to heat when metabolized. (One food calorie is 1000 calories and therefore equals 4186 J.) The body then moves all this heat to the surface by a combination of thermal conductivity and motion of the blood. The internal temperature of the body (where digestion occurs) is normally 37\(^\circ\)C, and the surface is usually about 30\(^\circ\)C. By how much do the digestion and metabolism of a 2.50-g pat of butter change your body's entropy? Does it increase or decrease?

As a budding mechanical engineer, you are called upon to design a Carnot engine that has 2.00 mol of a monatomic ideal gas as its working substance and operates from a high temperature reservoir at 500\(^\circ\)C. The engine is to lift a 15.0-kg weight 2.00 m per cycle, using 500 J of heat input. The gas in the engine chamber can have a minimum volume of 5.00 \(L\) during the cycle. (a) Draw a \(pV\)-diagram for this cycle. Show in your diagram where heat enters and leaves the gas. (b) What must be the temperature of the cold reservoir? (c) What is the thermal efficiency of the engine? (d) How much heat energy does this engine waste per cycle? (e) What is the maximum pressure that the gas chamber will have to withstand?

An experimental power plant at the Natural Energy Laboratory of Hawaii generates electricity from the temperature gradient of the ocean. The surface and deep-water temperatures are 27\(^\circ\)C and 6\(^\circ\)C, respectively. (a) What is the maximum theoretical efficiency of this power plant? (b) If the power plant is to produce 210 \(kW\) of power, at what rate must heat be extracted from the warm water? At what rate must heat be absorbed by the cold water? Assume the maximum theoretical efficiency. (c) The cold water that enters the plant leaves it at a temperature of 10\(^\circ\)C. What must be the flow rate of cold water through the system? Give your answer in kg/h and in \(L\)/h.

You decide to use your body as a Carnot heat engine. The operating gas is in a tube with one end in your mouth (where the temperature is 37.0\(^\circ\)C) and the other end at the surface of your skin, at 30.0\(^\circ\)C. (a) What is the maximum efficiency of such a heat engine? Would it be a very useful engine? (b) Suppose you want to use this human engine to lift a 2.50-kg box from the floor to a tabletop 1.20 m above the floor. How much must you increase the gravitational potential energy, and how much heat input is needed to accomplish this? (c) If your favorite candy bar has 350 food calories (1 food calorie = 4186 J) and 80% of the food energy goes into heat, how many of these candy bars must you eat to lift the box in this way?

A cylinder contains oxygen at a pressure of 2.00 atm. The volume is 4.00 \(L\), and the temperature is 300 \(K\). Assume that the oxygen may be treated as an ideal gas. The oxygen is carried through the following processes: (i) Heated at constant pressure from the initial state (state 1) to state 2, which has \(T\) = 450 K. (ii) Cooled at constant volume to 250 \(K\) (state 3). (iii) Compressed at constant temperature to a volume of 4.00 \(L\) (state 4). (iv) Heated at constant volume to 300 \(K\), which takes the system back to state 1. (a) Show these four processes in a \(pV\)-diagram, giving the numerical values of \(p\) and \(V\) in each of the four states. (b) Calculate \(Q\) and \(W\) for each of the four processes. (c) Calculate the net work done by the oxygen in the complete cycle. (d) What is the efficiency of this device as a heat engine? How does this compare to the efficiency of a Carnot cycle engine operating between the same minimum and maximum temperatures of 250 \(K\) and 450 \(K\)?

A Carnot engine operates between two heat reservoirs at temperatures \(T_H\) and \(T_C\) . An inventor proposes to increase the efficiency by running one engine between \(T_H\) and an intermediate temperature \(T'\) and a second engine between \(T'\) and \(T_C\) , using as input the heat expelled by the first engine. Compute the efficiency of this composite system, and compare it to that of the original engine.

A typical coal-fired power plant generates 1000 MW of usable power at an overall thermal efficiency of 40%. (a) What is the rate of heat input to the plant? (b) The plant burns anthracite coal, which has a heat of combustion of 2.65 \(\times\) 10\(^7\) J/kg. How much coal does the plant use per day, if it operates continuously? (c) At what rate is heat ejected into the cool reservoir, which is the nearby river? (d) The river is at 18.0\(^\circ\)C before it reaches the power plant and 18.5\(^\circ\)C after it has received the plant's waste heat. Calculate the river's flow rate, in cubic meters per second. (e) By how much does the river's entropy increase each second?

An air conditioner operates on 800 W of power and has a performance coefficient of 2.80 with a room temperature of 21.0\(^\circ\)C and an outside temperature of 35.0\(^\circ\)C. (a) Calculate the rate of heat removal for this unit. (b) Calculate the rate at which heat is discharged to the outside air. (c) Calculate the total entropy change in the room if the air conditioner runs for 1 hour. Calculate the total entropy change in the outside air for the same time period. (d) What is the net change in entropy for the system (room + outside air) ?

A person with skin of surface area 1.85 m\(^2\) and temperature 30.0\(^\circ\)C is resting in an insulated room where the ambient air temperature is 20.0\(^\circ\)C. In this state, a person gets rid of excess heat by radiation. By how much does the person change the entropy of the air in this room each second? (Recall that the room radiates back into the person and that the emissivity of the skin is 1.00.)

An object of mass \(m_1\), specific heat \(c_1\), and temperature \(T_1\) is placed in contact with a second object of mass \(m_2\), specific heat \(c_2\), and temperature \(T_2\) > \(T_1\). As a result, the temperature of the first object increases to \(T\) and the temperature of the second object decreases to \(T'\). (a) Show that the entropy increase of the system is $$\Delta S = m_1c_1 ln {T \over T_1} + m_2c_2 ln {T' \over T_2}$$ and show that energy conservation requires that $$m_1c_1 (T - T_1) = m_2c_2 (T_2 - T')$$ (b) Show that the entropy change \(\Delta S\), considered as a function of \(T\), is a \(maximum\) if \(T = T'\), which is just the condition of thermodynamic equilibrium. (c) Discuss the result of part (b) in terms of the idea of entropy as a measure of randomness.

For a refrigerator or air conditioner, the coefficient of performance \(K\) (often denoted as COP) is, as in Eq. (20.9), the ratio of cooling output \(Q_C\) 0 to the required electrical energy input \(W\) , both in joules. The coefficient of performance is also expressed as a ratio of powers, $$K = {(Q_C ) /t \over (W) /t}$$ where \(Q_C /t\) is the cooling power and \(W /t\) is the electrical power input to the device, both in watts. The energy efficiency ratio (\(EER\)) is the same quantity expressed in units of Btu for \(Q_C\) and \(W \cdot h\) for \(W\) . (a) Derive a general relationship that expresses \(EER\) in terms of \(K\). (b) For a home air conditioner, \(EER\) is generally determined for a 95\(^\circ\)F outside temperature and an 80\(^\circ\)F return air temperature. Calculate \(EER\) for a Carnot device that operates between 95\(^\circ\)F and 80\(^\circ\)F. (c) You have an air conditioner with an \(EER\) of 10.9. Your home on average requires a total cooling output of \(Q_C = 1.9 \times 10^{10} J\) per year. If electricity costs you 15.3 cents per \(kW \cdot h\), how much do you spend per year, on average, to operate your air conditioner? (Assume that the unit's \(EER\) accurately represents the operation of your air conditioner. A \(seasonal\) \(energy\) \(efficiency\) \(ratio\) (\(SEER\)) is often used. The \(SEER\) is calculated over a range of outside temperatures to get a more accurate seasonal average.) (d) You are considering replacing your air conditioner with a more efficient one with an \(EER\) of 14.6. Based on the \(EER\), how much would that save you on electricity costs in an average year?

Compare the entropy change of the warmer water to that of the colder water during one cycle of the heat engine, assuming an ideal Carnot cycle. (a) The entropy does not change during one cycle in either case. (b) The entropy of both increases, but the entropy of the colder water increases by more because its initial temperature is lower. (c) The entropy of the warmer water decreases by more than the entropy of the colder water increases, because some of the heat removed from the warmer water goes to the work done by the engine. (d) The entropy of the warmer water decreases by the same amount that the entropy of the colder water increases.

If the proposed plant is built and produces 10 \(MW\) but the rate at which waste heat is exhausted to the cold water is 165 \(MW\), what is the plant's actual efficiency? (a) 5.7%; (b) 6.1%; (c) 6.5%; (d) 16.5%.